Experimental aspects of the gyroscope's movement

Abstract

In presence of a uniform gravitational field, Euler equations for a gyroscope can be written as a non-linear equation for the components of Riemann's stereographic projection of the symmetry axis over a horizontal plane. Under the approximation of nutations with low amplitude, the solution of this equation corresponds to the sum of two rotating vectors with angular frequencies related to both angular velocities of nutation and precession. Such velocities are functions of rotation rapidity and inertia momentum of the gyroscope. From pictures of the movement projection of a commercial gyroscope, and using a laser that turn on during half revolution cycle of a disk, we can determine all kinematic quantities of the gyroscope, velocities of: rotation, precession and nutation, along with the angle of average inclination from axis. After complete a total of 120 experiments, we corroborate that the expressions given for velocities of precession and nutation, in function of rotation, match with experimental data. This is an easy experiment to implement, and can be used in advanced courses of mechanic.